Select Medium / माध्यम चुनें:

Eng (English) Beng (বাংলা) Hindi (हिन्दी)

WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 1

Estimated Time: 55 Mins

Study Progress: In Progress

Revision of Previous Lessons

Welcome to Chapter 1 "Revision of Previous Lessons" (পূর্বপাঠের পুনরাবৃত্তি) of the West Bengal Board of Secondary Education (WBBSE) Class 7 Mathematics (Ganit Prabha / গণিত প্রভা). Structured strictly according to the TargetExams Gold-Standard pedagogy, this master guide bridges upper primary arithmetic with middle-school analytical problem-solving. Students will master the rigorous VBODMAS operational hierarchy, multi-step fraction and decimal simplifications, LCM and HCF cardinal relationships, direct vs. inverse unitary method reasoning, and foundational perimeter, area, and angle properties.

Take Chapter CBT Test View All Chapters

Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

🌍 Have You Ever Wondered?

Why does $5 + 3 \times 2$ equal $11$, but on a simple pocket calculator it sometimes shows $16$?

If you give $5 + 3 \times 2$ to two different people, one might add first ($5 + 3 = 8$) and multiply by $2$ to get $16$. Another person might multiply first ($3 \times 2 = 6$) and add $5$ to get $11$. Both used standard arithmetic, so who is right?

Centuries ago, mathematicians, scientists, and navigators worldwide faced this exact dilemma. A rocket guidance computer or a bank transfer system cannot have two different answers for the same numbers! To prevent confusion and errors, global mathematicians established the inviolable Order of Operations (BODMAS/VBODMAS). Mathematics is not just about computing digits—it is a logical language with universal grammar rules that keep planes flying safely and bridges standing strong.

Why This Chapter Matters

Before You Begin (Prerequisites)

Basic understanding of natural numbers, whole numbers, and the decimal place value system.
Fluency in fundamental operations: addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$).
Concept of fractions as parts of a whole (numerator and denominator), and types of fractions (proper, improper, and mixed).

What You Will Learn (Core Objectives)

Apply the complete VBODMAS hierarchy (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction) to simplify complex algebraic and arithmetic expressions.
Execute multi-step fraction operations with reciprocals, simplification to lowest terms, and exact decimal conversions.
Master prime factorization and division methods to determine HCF and LCM, and verify the cardinal formula: $\text{First Number} \times \text{Second Number} = \text{HCF} \times \text{LCM}$.
Distinguish between Direct and Inverse Proportions and solve contextual real-life problems using the 2-step Unitary Method.
Compute the perimeter and area of rectangular and square regions, and calculate complementary and supplementary angle measures.

Chapter Roadmap & Progression

1 1. Order of Operations: The VBODMAS...

2 2. Operations with Fractions & Deci...

3 3. HCF, LCM & The Cardinal Number R...

4 4. The Unitary Method: Direct vs. I...

5 5. Foundational Mensuration & Angle...

Complete Concept Guide (100% Curriculum Coverage)

1. Order of Operations: The VBODMAS Rule & Bracket Hierarchy

1. The Intuition

When a mathematical expression contains multiple operations and nested brackets, evaluating them in random order creates chaos. In Class 7, we follow the universal grammar of mathematics called VBODMAS to guarantee one single, correct solution.

2. The Formal Concept & Structure

Expressions must be evaluated strictly according to the hierarchical priority table from left to right:

Letter	Full Form	Symbol & Meaning	Priority Level
V	Vinculum (Bar Bracket)	$\overline{\text{bar}}$ (Bar above expression)	1st (Evaluated first)
B	Brackets	Round $( )$, Curly $\{ \}$, Square $[ ]$	2nd (Innermost to outermost)
O	Of	'Of' (Implied multiplication)	3rd (Before division)
D	Division	$\div$ (Equal rank with multiplication)	4th (Left-to-right)
M	Multiplication	$\times$ (Equal rank with division)	5th (Left-to-right)
A	Addition	$+$ (Equal rank with subtraction)	6th (Left-to-right)
S	Subtraction	$-$ (Final resolving operation)	7th (Left-to-right)

Golden Left-to-Right Tie-Breaker: When operations of equal rank appear side-by-side (such as Division and Multiplication, or Addition and Subtraction), always evaluate strictly from left to right.

3. Concrete Worked Example

Example: Simplify the nested bracket expression: $36 - [18 - \{14 - (15 - \overline{4 - 2})\}]$

Step 1 (Resolve Vinculum): $\overline{4 - 2} = 2$

Expression becomes: $36 - [18 - \{14 - (15 - 2)\}]$

Step 2 (Evaluate Parentheses): $15 - 2 = 13$

Expression becomes: $36 - [18 - \{14 - 13\}]$

Step 3 (Evaluate Braces): $14 - 13 = 1$

Expression becomes: $36 - [18 - 1]$

Step 4 (Evaluate Square Brackets): $18 - 1 = 17$

Step 5 (Final Subtraction): $36 - 17 = 19$

Answer: $\mathbf{19}$

4. Pitfall & Examiner Trap

⚠️ Trap: The Vinculum Negative Sign Error
In an expression like $-(5 - \overline{3 - 1})$, students often mistakenly distribute the minus sign before resolving the bar, writing $-3 - 1 = -4$.
Rule: The vinculum acts as an independent grouping bracket. Always calculate the expression under the bar ($3 - 1 = 2$) first, and then apply the outer negative sign: $-(5 - 2) = -3$.

5. Why This Matters in Life

Every computer compiler (Python, JavaScript, C++) and financial spreadsheet (Excel) implements this exact operator precedence parser to ensure multi-billion-dollar transactions are processed without error.

2. Operations with Fractions & Decimal Place Values

1. The Intuition

Why does dividing by $\frac{1}{2}$ double your answer? Imagine you have $6$ whole cakes, and you cut each cake into half-portions ($\frac{1}{2}$). How many slices do you get? $6 \div \frac{1}{2} = 6 \times 2 = 12$ slices! Dividing by any fraction is mathematically identical to multiplying by its reciprocal (inverted fraction).

2. The Formal Concept & Rules

Key Fractional Operations:

Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ (Multiply numerators together, multiply denominators together).
Division by Reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$ (Invert the divisor and multiply).
Converting to Terminating Decimals: Convert the denominator to powers of 10 ($10, 100, 1000$). For instance: $\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35$.

3. Concrete Worked Example

Example: Simplify: $\frac{3}{4} \div \frac{9}{16} \text{ of } \frac{2}{3} + \frac{5}{6}$

Step 1 ('Of' Operation First): $\frac{9}{16} \text{ of } \frac{2}{3} = \frac{9 \times 2}{16 \times 3} = \frac{18}{48} = \frac{3}{8}$

Expression becomes: $\frac{3}{4} \div \frac{3}{8} + \frac{5}{6}$

Step 2 (Division via Reciprocal): $\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \times \frac{8}{3} = 2$

Step 3 (Addition): $2 + \frac{5}{6} = \frac{12 + 5}{6} = \frac{17}{6} = 2\frac{5}{6}$

Answer: $\mathbf{2\frac{5}{6}}$ or $\mathbf{\frac{17}{6}}$

4. Pitfall & Examiner Trap

⚠️ Trap: Dividing before "Of"
In $\frac{3}{4} \div \frac{9}{16} \text{ of } \frac{2}{3}$, students often calculate the division first. Remember: in VBODMAS, "O" (Of) takes absolute precedence over "D" (Division)!

5. Why This Matters in Life

Mastering fraction scaling is essential for architectural drafting (1:50 blueprint ratios), culinary recipes, and medical dosage dilutions.

3. HCF, LCM & The Cardinal Number Relationship

1. The Intuition

Suppose three bus routes depart from the same terminal every 12 minutes, 15 minutes, and 18 minutes. When will all three buses depart simultaneously again? Finding the Lowest Common Multiple (LCM) answers this exact scheduling problem ($180$ minutes = 3 hours).

2. The Cardinal Product Formula

For any two positive natural numbers $a$ and $b$:

$$\mathbf{\text{Product of Numbers } (a \times b) = \text{HCF}(a, b) \times \text{LCM}(a, b)}$$

Therefore, if either the HCF or LCM and one number is known, the missing quantity can be found instantly:

$$\text{Second Number} = \frac{\text{HCF} \times \text{LCM}}{\text{First Number}}$$

HCF & LCM of Fractions:

$$\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

$$\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

3. Concrete Worked Example

Example: The HCF of two numbers is $8$ and their LCM is $280$. If one number is $56$, find the other number. Also, calculate the LCM of fractions $\frac{2}{3}, \frac{8}{9}, \frac{16}{81}$.

Part 1 (Number Relation):
$\text{Other Number} = \frac{\text{HCF} \times \text{LCM}}{\text{Given Number}} = \frac{8 \times 280}{56} = \frac{2240}{56} = \mathbf{40}$

Part 2 (Fractional LCM):
LCM of numerators $(2, 8, 16) = 16$
HCF of denominators $(3, 9, 81) = 3$
$\text{LCM of Fractions} = \frac{16}{3} = \mathbf{5\frac{1}{3}}$

Answers: Number $= \mathbf{40}$, Fraction LCM $= \mathbf{5\frac{1}{3}}$

4. Pitfall & Examiner Trap

⚠️ The Three-Number Misconception:
The formula $a \times b = \text{HCF} \times \text{LCM}$ is valid ONLY for two numbers! For three numbers $a, b, c$, the product $a \times b \times c \ne \text{HCF} \times \text{LCM}$. Never use this formula for three or more numbers!

5. Why This Matters in Life

Logistics route planners, telecommunications packet dispatchers, and railway scheduling systems rely directly on LCM and HCF algorithms to optimize throughput.

4. The Unitary Method: Direct vs. Inverse Proportions

1. The Intuition

The Unitary Method means "finding the value of one unit first". But relationship direction matters: buying more books costs more money (Direct Variation). But assigning more workers to dig a canal takes less time (Inverse Variation)! Understanding the underlying physics prevents costly calculation errors.

2. The Formal Comparison Structure

Property	Direct Variation	Inverse Variation
Behavior	Both quantities increase or decrease together ($\frac{y}{x} = k$)	One quantity increases as the other decreases ($x \times y = k$)
Real Examples	Items and cost; Distance and fuel consumption	Workers and completion days; Speed and travel time
Finding Unit Value	Divide to find the value of 1 unit	Multiply to find the value of 1 unit

3. Concrete Worked Example

Example: A military camp of 120 soldiers had ration provisions for 30 days. After 10 days, 30 soldiers were transferred out. How many days will the remaining rations last for the remaining soldiers?

Step 1 (Remaining Quantities):
Remaining days for 120 soldiers $= 30 - 10 = 20\text{ days}$.
Remaining soldiers $= 120 - 30 = 90\text{ soldiers}$.

Step 2 (Inverse Unitary Formulation):
Food lasts for 120 soldiers $= 20\text{ days}$
Food lasts for 1 soldier $= 20 \times 120 = 2400\text{ days}$ (fewer soldiers $\to$ lasts longer $\to$ Multiply)
Food lasts for 90 soldiers $= \frac{2400}{90} = \mathbf{26\frac{2}{3}\text{ days}}$

Answer: The remaining rations will last for $\mathbf{26\frac{2}{3}\text{ days}}$.

4. Pitfall & Examiner Trap

⚠️ The Division Trap in Inverse Variation:
Students automatically divide in step 1 ($20 \div 120$), mistakenly assuming 1 soldier takes a fraction of a day. Always verify physical sense: one single person will take vastly longer to consume an entire warehouse of food!

5. Why This Matters in Life

Civil engineers, construction contractors, and software sprint managers continuously use inverse unitary models to schedule workforce sizes and prevent deadline delays.

5. Foundational Mensuration & Angle Relationships

1. The Intuition

Perimeter is a 1-dimensional boundary measurement (like walking along a fence around a park), whereas Area is a 2-dimensional surface measurement (like mowing the entire grass inside the park). Confusing the two leads to ordering the wrong building materials.

2. The Formal Concept & Formulas

Key Mensuration Formulas:

Rectangle: $\text{Perimeter} = 2(l + b)$ units; $\text{Area} = l \times b$ square units.
Square: $\text{Perimeter} = 4s$ units; $\text{Area} = s^2$ square units.

Angle Relationships:

Complementary Angles: Two angles whose sum is exactly $90^\circ$ ($\angle 1 + \angle 2 = 90^\circ$).
Supplementary Angles: Two angles whose sum is exactly $180^\circ$ ($\angle 1 + \angle 2 = 180^\circ$).

3. Concrete Worked Example

Example: A hall is $12\text{ m}$ long and $8\text{ m}$ wide. Calculate the total cost to tile the floor at ₹$45/\text{m}^2$ and install boundary wooden skirting at ₹$15/\text{m}$. Also, find an angle which is $30^\circ$ less than its supplement.

Part 1 (Flooring & Skirting):
Floor Area $= 12 \times 8 = 96\text{ m}^2 \implies \text{Tiling Cost} = 96 \times 45 = \mathbf{₹4,320}$
Perimeter $= 2(12 + 8) = 40\text{ m} \implies \text{Skirting Cost} = 40 \times 15 = \mathbf{₹600}$
Total Project Cost $= 4320 + 600 = \mathbf{₹4,920}$

Part 2 (Angle Calculation):
Let angle be $x$. Its supplement is $180^\circ - x$.
Equation: $x = (180^\circ - x) - 30^\circ \implies 2x = 150^\circ \implies x = \mathbf{75^\circ}$.

Answers: Total Cost $= \mathbf{₹4,920}$, Angle $= \mathbf{75^\circ}$.

4. Pitfall & Examiner Trap

⚠️ Unit Dimensionality Trap:
Writing area as $\text{m}$ instead of $\text{m}^2$, or forgetting to multiply by 2 for rectangle perimeter ($l + b$ instead of $2(l+b)$). Always check units carefully!

5. Why This Matters in Life

Interior designers calculating flooring tiles, solar panel engineers maximizing roof surface areas, and property surveyors all utilize these principles daily.

Key Formulas, Identities & Theorems

VBODMAS Operational Hierarchy

$$\text{Vinculum} \rightarrow \text{Brackets} \rightarrow \text{Of} \rightarrow \text{Division} \rightarrow \text{Multiplication} \rightarrow \text{Addition} \rightarrow \text{Subtraction}$$

Universal precedence standard

Cardinal Product Rule

$$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$

Valid strictly for any two positive integers

LCM of Fractions

$$\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

Standard fractional LCM formula

HCF of Fractions

$$\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

Standard fractional HCF formula

Perimeter & Area of Rectangle

$$\text{Perimeter} = 2(l + b), \quad \text{Area} = l \times b$$

l = length, b = breadth

Complementary & Supplementary Angles

$$\angle 1 + \angle 2 = 90^\circ \text{ (Comp)}, \quad \angle 1 + \angle 2 = 180^\circ \text{ (Supp)}$$

Pairwise angular geometric relationships

Conceptual Solved Examples & Case Studies

Example 1

Simplify the nested expression: $16 \div [4 + \{8 - (6 - \overline{4 - 2})\}]$

Step-by-Step Solution:

Step 1 (Vinculum): $\overline{4 - 2} = 2 \implies 16 \div [4 + \{8 - (6 - 2)\}]$
Step 2 (Parentheses): $6 - 2 = 4 \implies 16 \div [4 + \{8 - 4\}]$
Step 3 (Braces): $8 - 4 = 4 \implies 16 \div [4 + 4]$
Step 4 (Square Brackets): $4 + 4 = 8$
Step 5 (Final Division): $16 \div 8 = \mathbf{2}$
Answer: $\mathbf{2}$

Example 2

The HCF of two numbers is $8$ and their LCM is $280$. If one number is $56$, find the other number.

Step-by-Step Solution:

Using the Cardinal Formula: $\text{First} \times \text{Second} = \text{HCF} \times \text{LCM}$
$$\text{Other Number} = \frac{8 \times 280}{56} = \frac{2240}{56} = \mathbf{40}$$
Answer: The other number is $\mathbf{40}$.

Example 3

Find an angle whose measure is $30^\circ$ less than its supplement.

Step-by-Step Solution:

Let the angle be $x$. Its supplement is $(180^\circ - x)$.
According to question: $x = (180^\circ - x) - 30^\circ$
$$x + x = 150^\circ \implies 2x = 150^\circ \implies x = \mathbf{75^\circ}$$
Check: Supplement of $75^\circ$ is $105^\circ$, and $105^\circ - 30^\circ = 75^\circ$.
Answer: The angle is $\mathbf{75^\circ}$.

Common Misconceptions & Examiner Traps

Common Misconception

Distributing the negative sign into a vinculum before computing the bar value.

Scientific Reality & Correction

Always evaluate the operation under the vinculum independently first, then apply outer signs.

Common Misconception

Performing division before 'Of'.

Scientific Reality & Correction

In VBODMAS, 'Of' represents high-priority multiplication that must be resolved prior to division.

Common Misconception

Applying the Product Formula $\text{Product} = \text{HCF} \times \text{LCM}$ to three numbers.

Scientific Reality & Correction

This relationship strictly holds for two numbers only. For three numbers, factorize each separately.

Common Misconception

Confusing linear perimeter units with square area units.

Scientific Reality & Correction

Perimeter is measured in linear units ($\text{cm}, \text{m}$), whereas Area is measured in square units ($\text{cm}^2, \text{m}^2$).

Visual Learning & Conceptual Map

WBBSE Mathematics Foundations: The Operator Hierarchy & Number Operations Compass

Always evaluate from left to right strictly according to this precedence pipeline

1. V

Vinculum $\overline{x}$

Highest Rank

2. B

Brackets $( ) \{ \} [ ]$

Inside → Out

3. O

'Of' (এর)

Pre-multiply

4. D

Division $\div$

Left → Right

5. M

Multiply $\times$

Equal to Div

6. A

Add $+$

Equal to Sub

7. S

Subtract $-$

Final Resolution

Chapter Summary & 10 Key Takeaways

Takeaway 1

VBODMAS Precedence: Strictly evaluate in order: Vinculum $\rightarrow$ Brackets $\rightarrow$ Of $\rightarrow$ Division $\rightarrow$ Multiplication $\rightarrow$ Addition $\rightarrow$ Subtraction.

Takeaway 2

Fractional Division: Dividing by a fraction is identical to multiplying by its reciprocal (inverted form $\frac{d}{c}$).

Takeaway 3

Cardinal Relationship: For any two positive natural numbers, $\text{First Number} \times \text{Second Number} = \text{HCF} \times \text{LCM}$.

Takeaway 4

Fractional HCF/LCM: $\text{LCM} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$ and $\text{HCF} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$.

Takeaway 5

Direct Proportion: Quantities increase together proportionally; divide to find 1 unit value.

Takeaway 6

Inverse Proportion: One quantity increases as the other decreases proportionally; multiply to find 1 unit value.

Takeaway 7

Angles: Complementary angles sum to $90^\circ$, Supplementary angles sum to $180^\circ$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Simplify the expression: $20 - [5 + \{10 - (8 - \overline{5 - 2})\}]$

Reveal Answer & Explanation

Answer: $20 - [5 + \{10 - (8 - 3)\}] = 20 - [5 + \{10 - 5\}] = 20 - [5 + 5] = 20 - 10 = \mathbf{10}$
Resolve $\overline{5 - 2} = 3$ first, followed by round brackets, curly braces, and square brackets in sequence.

Find the HCF of the fractions $\frac{3}{5}, \frac{6}{25}, \frac{9}{10}$.

Reveal Answer & Explanation

Answer: $\mathbf{\frac{3}{50}}$ — HCF of numerators $(3, 6, 9) = 3$; LCM of denominators $(5, 25, 10) = 50$. Formula: $\frac{\text{HCF}}{\text{LCM}} = \frac{3}{50}$.
Use the formula: $\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$.

The HCF of two numbers is $12$ and their LCM is $144$. If one number is $36$, find the other number.

Reveal Answer & Explanation

Answer: $\text{Other Number} = \frac{12 \times 144}{36} = \mathbf{48}$
Apply the cardinal formula: $\text{Second Number} = (\text{HCF} \times \text{LCM}) / \text{First Number}$.

8 workers can finish a construction task in 15 days. How many days will 10 workers take to complete the same task?

Reveal Answer & Explanation

Answer: 8 workers take 15 days $\implies$ 1 worker takes $15 \times 8 = 120$ days $\implies$ 10 workers take $\frac{120}{10} = \mathbf{12\text{ days}}$.
This is an inverse proportion problem: more workers take fewer days.

Find an angle whose measure is $40^\circ$ less than its supplement.

Reveal Answer & Explanation

Answer: Let angle be $x$. $x = (180^\circ - x) - 40^\circ \implies 2x = 140^\circ \implies x = \mathbf{70^\circ}$.
The sum of supplementary angles is $180^\circ$. Set up the linear equation $x + (x + 40^\circ) = 180^\circ$.

Finished Studying This Chapter?

READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test

15 MCQs • Core Concepts

Advanced Test

15 MCQs • HOTS & Traps